By K. Borsuk [1]}, for \(n\geqslant 2\)  there exists a
Cantor set in \(\mathbb {R}^n\)  such that its projection to every hyperplane contains a \((n-1)\) -dimensional ball, equivalently, has
dimension \((n-1)\) . (For \(n=3\) ,
see an alternative description in
[2]};
another proof of Borsuk's result,
using universal surjectivity property of \(\mathcal {C}\) , can be found in [3]}.)
As a corollary,
the projection of the Borsuk set to each
\(m\) -plane, \(m\leqslant n-1\) ,
has dimension \(m\) .
We remark that
a set with this property can
be obtained from any
given Cantor set, using a small
ambient isotopy (Statement REF ).
For \(n\geqslant 3\) , in \(\mathbb {R}^n\) 
there is no Cantor set
all of whose projections to \((n-1)\) -planes
are convex bodies
[4]};
generalizations are obtained in
[3]},
[6]}.
In \(\mathbb {R}^3\) , we construct a convex body \(K\) 
and a Cantor set \(\mathcal {A}\subset K\) 
such that the projections of \(K\) 
and \(\mathcal {A}\)  into each line coincide
(Corollary REF ).
